Complete these steps 3 times with mean vector = c(0,0,0), and
differenct covariance matrices each time. Select values for your
covariance matrices. Each time use 3-dimensions, X, Y, and Z.
1st Time
Multivariate normal density
Create a grid of values going from -3, to 3 for each dimension. Use
expand.grid() to get the grid of values.
Compute the density of each point.
Plot a scatterplot matrix showing the density of points.
library(mvtnorm)
v <- seq(
from = -3,
to = 3,
length = 20
)
M_multivariatenormal <- expand.grid(
x = v,
y = v,
z = v
)
v_mean <- c(0,0,0)
M_sigma <- matrix(
data = c(
2,-1,0,
-1,2,1,
0,1,2
),
nrow = 3,
byrow = TRUE
)
M_multivariatenormal$density <- mvtnorm::dmvnorm(
x = M_multivariatenormal,
mean = v_mean,
sigma = M_sigma
)
M_multivariatenormal$density_bucket <- ggplot2::cut_number(
x = M_multivariatenormal$density,
n = 4
)
v_color <- rev(gray.colors(
n = length(levels(M_multivariatenormal$density_bucket)),
start = 0,
end = 1
))
names(v_color) <- levels(M_multivariatenormal$density_bucket)
M_multivariatenormal$color <- v_color[as.character(M_multivariatenormal$density_bucket)]
pairs(
x = M_multivariatenormal[,c("x","y","z")],
col = M_multivariatenormal$color,
pch = 19,
main = "Scatterplot matrix of the multivariate normal density 1"
)
Multivariate normal random number generation.
Randomly generate at least 100 multivariate normal points.
Plot a 3-d scatterplot of your points.
M_rnvnorm <- mvtnorm::rmvnorm(
n = 100,
mean = v_mean,
sigma = M_sigma
)
colnames(M_rnvnorm) <- c("x","y","z")
M_rnvnorm <- as.data.frame(
x = M_rnvnorm
)
p <- plotly::plot_ly(
x = M_rnvnorm$x,
y = M_rnvnorm$y,
z = M_rnvnorm$z,
type="scatter3d",
mode="markers"
)
p
Multivariate probabilities
Compute the probability that an observation is in this region:
mvtnorm::pmvnorm(
lower = c(-1,-2,-2),
upper = c(2,1,3),
mean = v_mean,
sigma =M_sigma
)
## [1] 0.4652876
## attr(,"error")
## [1] 0.000116656
## attr(,"msg")
## [1] "Normal Completion"
2nd Try
Multivariate normal density
Create a grid of values going from -3, to 3 for each dimension. Use
expand.grid() to get the grid of values.
Compute the density of each point.
Plot a scatterplot matrix showing the density of points.
library(mvtnorm)
v <- seq(
from = -3,
to = 3,
length = 20
)
M_multivariatenormal <- expand.grid(
x = v,
y = v,
z = v
)
v_mean <- c(0,0,0)
M_sigma <- matrix(
data = c(
1,-1,0,
-1,2,1,
0,1,3
),
nrow = 3,
byrow = TRUE
)
M_multivariatenormal$density <- mvtnorm::dmvnorm(
x = M_multivariatenormal,
mean = v_mean,
sigma = M_sigma
)
M_multivariatenormal$density_bucket <- ggplot2::cut_number(
x = M_multivariatenormal$density,
n = 4
)
v_color <- rev(gray.colors(
n = length(levels(M_multivariatenormal$density_bucket)),
start = 0,
end = 1
))
names(v_color) <- levels(M_multivariatenormal$density_bucket)
M_multivariatenormal$color <- v_color[as.character(M_multivariatenormal$density_bucket)]
pairs(
x = M_multivariatenormal[,c("x","y","z")],
col = M_multivariatenormal$color,
pch = 19,
main = "Scatterplot matrix of the multivariate normal density 2 "
)
Multivariate normal random number generation.
Randomly generate at least 100 multivariate normal points.
Plot a 3-d scatterplot of your points.
M_rnvnorm <- mvtnorm::rmvnorm(
n = 100,
mean = v_mean,
sigma = M_sigma
)
colnames(M_rnvnorm) <- c("x","y","z")
M_rnvnorm <- as.data.frame(
x = M_rnvnorm
)
p <- plotly::plot_ly(
x = M_rnvnorm$x,
y = M_rnvnorm$y,
z = M_rnvnorm$z,
type="scatter3d",
mode="markers"
)
p
Multivariate probabilities
Compute the probability that an observation is in this region:
mvtnorm::pmvnorm(
lower = c(-1,-2,-2),
upper = c(2,1,3),
mean = v_mean,
sigma =M_sigma
)
## [1] 0.5304862
## attr(,"error")
## [1] 0.0005668327
## attr(,"msg")
## [1] "Normal Completion"
3rd Try
Multivariate normal density
Create a grid of values going from -3, to 3 for each dimension. Use
expand.grid() to get the grid of values.
Compute the density of each point.
Plot a scatterplot matrix showing the density of points.
library(mvtnorm)
v <- seq(
from = -3,
to = 3,
length = 20
)
M_multivariatenormal <- expand.grid(
x = v,
y = v,
z = v
)
v_mean <- c(0,0,0)
M_sigma <- matrix(
data = c(
6,-1,0,
-1,5,1,
0,1,4
),
nrow = 3,
byrow = TRUE
)
M_multivariatenormal$density <- mvtnorm::dmvnorm(
x = M_multivariatenormal,
mean = v_mean,
sigma = M_sigma
)
M_multivariatenormal$density_bucket <- ggplot2::cut_number(
x = M_multivariatenormal$density,
n = 4
)
v_color <- rev(gray.colors(
n = length(levels(M_multivariatenormal$density_bucket)),
start = 0,
end = 1
))
names(v_color) <- levels(M_multivariatenormal$density_bucket)
M_multivariatenormal$color <- v_color[as.character(M_multivariatenormal$density_bucket)]
pairs(
x = M_multivariatenormal[,c("x","y","z")],
col = M_multivariatenormal$color,
pch = 19,
main = "Scatterplot matrix of the multivariate normal density"
)
Multivariate normal random number generation.
Randomly generate at least 100 multivariate normal points.
Plot a 3-d scatterplot of your points.
M_rnvnorm <- mvtnorm::rmvnorm(
n = 100,
mean = v_mean,
sigma = M_sigma
)
colnames(M_rnvnorm) <- c("x","y","z")
M_rnvnorm <- as.data.frame(
x = M_rnvnorm
)
p <- plotly::plot_ly(
x = M_rnvnorm$x,
y = M_rnvnorm$y,
z = M_rnvnorm$z,
type="scatter3d",
mode="markers"
)
p
Multivariate probabilities
Compute the probability that an observation is in this region:
mvtnorm::pmvnorm(
lower = c(-1,-2,-2),
upper = c(2,1,3),
mean = v_mean,
sigma =M_sigma
)
## [1] 0.17455
## attr(,"error")
## [1] 1.095076e-06
## attr(,"msg")
## [1] "Normal Completion"